The difference app service types are:

Logic apps: Offers us a place where we can create a complex work flow to automate a business process. In a logic apps work flow we can call api that live any where. They host and execute these work flows

Mobile apps:We can host a back-end for our mobile app in IOS or Android devices. Provides unique capabilities of running the mobile client to work offline and later sync the changes. Offers the ability for push notifications

Function apps:

Provide a way of running an independent line of code that can respond to an event. The code is not part of the website or another application

Note: All the app services run on top of the Azure service fabric which provides them with resources